Groupe d’études et de recherche en analyse des décisions

Gradient proximal stochastique et applications pour l'inférence de modèles à effets mixtes en grande dimension

Éric Moulines Télécom ParisTech, France

We study a perturbed version of the proximal gradient algorithm for which the gradient is not known in closed form and should be approximated. We address the convergence and derive a non-asymptotic bound on the convergence rate for the perturbed proximal gradient, a perturbed averaged version of the proximal gradient algorithm and a perturbed version of the fast iterative shrinkage-thresholding (FISTA) of Beck and Teboulle (2009). When the approximation is achieved by using Monte Carlo methods, we derive conditions involving the Monte Carlo batch-size and the step-size of the algorithm under which convergence is guaranteed. In particular, we show that the Monte Carlo approximations of some averaged proximal gradient algorithms and a Monte Carlo approximation of FISTA achieve the same convergence rates as their deterministic counterparts. To illustrate, we apply the algorithms to high-dimensional generalized linear mixed models using \(\ell_1\)-penalization.